{"?xml":{"@version":"1.0"},"edm:RDF":{"@xmlns:dc":"http://purl.org/dc/elements/1.1/","@xmlns:edm":"http://www.europeana.eu/schemas/edm/","@xmlns:wgs84_pos":"http://www.w3.org/2003/01/geo/wgs84_pos","@xmlns:foaf":"http://xmlns.com/foaf/0.1/","@xmlns:rdaGr2":"http://rdvocab.info/ElementsGr2","@xmlns:oai":"http://www.openarchives.org/OAI/2.0/","@xmlns:owl":"http://www.w3.org/2002/07/owl#","@xmlns:rdf":"http://www.w3.org/1999/02/22-rdf-syntax-ns#","@xmlns:ore":"http://www.openarchives.org/ore/terms/","@xmlns:skos":"http://www.w3.org/2004/02/skos/core#","@xmlns:dcterms":"http://purl.org/dc/terms/","edm:WebResource":[{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:doc-6VFZ6PM1/72cf432c-f0e6-4df7-a8f2-df7abe41ede2/PDF","dcterms:extent":"402 KB"},{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:doc-6VFZ6PM1/63620947-3cdc-4cd4-b00f-a81b51427bfb/TEXT","dcterms:extent":"22 KB"}],"edm:TimeSpan":{"@rdf:about":"2008-2025","edm:begin":{"@xml:lang":"en","#text":"2008"},"edm:end":{"@xml:lang":"en","#text":"2025"}},"edm:ProvidedCHO":{"@rdf:about":"URN:NBN:SI:doc-6VFZ6PM1","dcterms:isPartOf":[{"@rdf:resource":"https://www.dlib.si/details/URN:NBN:SI:spr-UP1WMFAR"},{"@xml:lang":"sl","#text":"Ars mathematica contemporanea"}],"dcterms:issued":"2020","dc:creator":"Nakamigawa, Tomoki","dc:format":[{"@xml:lang":"sl","#text":"letnik:18"},{"@xml:lang":"sl","#text":"številka:2"},{"@xml:lang":"sl","#text":"str. 381-391"}],"dc:identifier":["ISSN:1855-3966","COBISSID_HOST:42357763","URN:URN:NBN:SI:doc-6VFZ6PM1"],"dc:language":"en","dc:publisher":{"@xml:lang":"sl","#text":"Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije"},"dc:subject":[{"@xml:lang":"en","#text":"chord diagram"},{"@xml:lang":"en","#text":"chord expansion"},{"@xml:lang":"en","#text":"Genocchi number"},{"@xml:lang":"sl","#text":"Genocchijevo število"},{"@xml:lang":"en","#text":"Seidel triangle"},{"@xml:lang":"sl","#text":"Seidelov trikotnik"},{"@xml:lang":"sl","#text":"tetivni diagram"},{"@xml:lang":"sl","#text":"tetivni razplet"}],"dcterms:temporal":{"@rdf:resource":"2008-2025"},"dc:title":{"@xml:lang":"sl","#text":"The expansion of a chord diagram and the Genocchi numbers|"},"dc:description":[{"@xml:lang":"sl","#text":"A chord diagram ?$E$? is a set of chords of a circle such that no pair of chords has a common endvertex. Let ?$v_1, v_2, \\dots, v_{2n}$? be a sequence of vertices arranged in clockwise order along a circumference. A chord diagram ?$\\{v_1v_{n+1}, v_2v_{n+2}, \\dots, v_nv_{2n}\\}$? is called an ?$n$?-crossing and a chord diagram ?$\\{v_1v_2, v_3v_4, \\dots, v_{2n-1}v_{2n}\\}$? is called an ?$n$?-necklace. For a chord diagram ?$E$? having a 2-crossing ?$S = \\{x_1x_3, x_2x_4\\}$?, the expansion of ?$E$? with respect to ?$S$? is to replace ?$E$? with ?$E_1 = (E \\setminus S) \\cup \\{x_2x_3, x_4x_1\\}$? or ?$E_2 = (E \\setminus S) \\cup \\{x_1x_2, x_3x_4\\}$?. Beginning from a given chord diagram ?$E$? as the root, by iterating chord expansions in both ways, we have a binary tree whose all leaves are nonintersecting chord diagrams. Let ?$\\mathcal{NCD}(E)$? be the multiset of the leaves. In this paper, the multiplicity of an ?$n$?-necklace in ?$\\mathcal{NCD}(E)$? is studied. Among other results, it is shown that the multiplicity of an ?$n$?-necklace generated from an ?$n$?-crossing equals the Genocchi number when ?$n$? is odd and the median Genocchi number when ?$n$? is even"},{"@xml:lang":"sl","#text":"Tetivni diagram ?$E$? je množica tetiv kroga, v kateri noben par tetiv nima skupnega krajišča. Naj bo ?$v_1, v_2, \\dots, v_{2n}$? zaporedje točk, urejenih v smeri urinega kazalca vzdolž oboda kroga. Tetivni diagram ?$\\{v_1v_{n+1}, v_2v_{n+2}, \\dots, v_nv_{2n}\\}$? se imenuje ?$n$?-križišče, tetivni diagram ?$\\{v_1v_2, v_3v_4, \\dots, v_{2n-1}v_{2n}\\}$? pa je ?$n$?-ogrlica. Naj bo ?$E$? tetivni diagram, ki ima 2-križišče ?$S = \\{x_1x_3, x_2x_4\\}$?; potem se zamenjava ?$E$? z ?$E_1 = (E \\setminus S) \\cup \\{x_2x_3, x_4x_1\\}$? ali z ?$E_2 = (E \\setminus S) \\cup \\{x_1x_2, x_3x_4\\}$? imenuje razplet ?$E$? glede na ?$S$?. Če začnemo z danim tetivnim diagramom ?$E$? kot korenom, potem pa delamo tetivne razplete na oba načina, dobimo dvojiško drevo, katerega listi so izključno tetivni diagrami brez križišč. Naj bo ?$\\mathcal{NCD}(E)$? mnogotera množica listov tega drevesa. V tem članku preučujemo večkratnost ?$n$?-ogrlice v mnogoteri množici ?$\\mathcal{NCD}(E)$?. Poleg drugih rezultatov, ki jih dobimo, pokažemo tudi, da je večkratnost ?$n$?-ogrlice, dobljene iz ?$n$?-križišča, enaka Genocchijevemu številu, če je ?$n$? liho število, in sredinskemu Genocchijevemu številu, če je ?$n$? sodo število"}],"edm:type":"TEXT","dc:type":[{"@xml:lang":"sl","#text":"znanstveno časopisje"},{"@xml:lang":"en","#text":"journals"},{"@rdf:resource":"http://www.wikidata.org/entity/Q361785"}]},"ore:Aggregation":{"@rdf:about":"http://www.dlib.si/?URN=URN:NBN:SI:doc-6VFZ6PM1","edm:aggregatedCHO":{"@rdf:resource":"URN:NBN:SI:doc-6VFZ6PM1"},"edm:isShownBy":{"@rdf:resource":"http://www.dlib.si/stream/URN:NBN:SI:doc-6VFZ6PM1/72cf432c-f0e6-4df7-a8f2-df7abe41ede2/PDF"},"edm:rights":{"@rdf:resource":"http://creativecommons.org/licenses/by/4.0/"},"edm:provider":"Slovenian National E-content Aggregator","edm:intermediateProvider":{"@xml:lang":"en","#text":"National and University Library of Slovenia"},"edm:dataProvider":{"@xml:lang":"sl","#text":"Univerza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije"},"edm:object":{"@rdf:resource":"http://www.dlib.si/streamdb/URN:NBN:SI:doc-6VFZ6PM1/maxi/edm"},"edm:isShownAt":{"@rdf:resource":"http://www.dlib.si/details/URN:NBN:SI:doc-6VFZ6PM1"}}}}